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A373673 First element of each maximal run of powers of primes (including 1).

%I A373673 #9 Jun 17 2024 08:51:43
%S A373673 1,7,11,13,16,19,23,25,27,29,31,37,41,43,47,49,53,59,61,64,67,71,73,
%T A373673 79,81,83,89,97,101,103,107,109,113,121,125,127,131,137,139,149,151,
%U A373673 157,163,167,169,173,179,181,191,193,197,199,211,223,227,229,233,239
%N A373673 First element of each maximal run of powers of primes (including 1).
%C A373673 A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.
%C A373673 The last element of the same run is A373674.
%C A373673 Consists of all powers of primes k such that k-1 is not a power of primes.
%H A373673 Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.
%e A373673 The maximal runs of powers of primes begin:
%e A373673    1   2   3   4   5
%e A373673    7   8   9
%e A373673   11
%e A373673   13
%e A373673   16  17
%e A373673   19
%e A373673   23
%e A373673   25
%e A373673   27
%e A373673   29
%e A373673   31  32
%e A373673   37
%e A373673   41
%e A373673   43
%e A373673   47
%e A373673   49
%t A373673 pripow[n_]:=n==1||PrimePowerQ[n];
%t A373673 Min/@Split[Select[Range[100],pripow],#1+1==#2&]//Most
%Y A373673 For composite antiruns we have A005381, max A068780, length A373403.
%Y A373673 For prime antiruns we have A006512, max A001359, length A027833.
%Y A373673 For composite runs we have A008864, max A006093, length A176246.
%Y A373673 For prime runs we have A025584, max A067774, length A251092 or A175632.
%Y A373673 For runs of prime-powers:
%Y A373673 - length A174965
%Y A373673 - min A373673 (this sequence)
%Y A373673 - max A373674
%Y A373673 - sum A373675
%Y A373673 For runs of non-prime-powers:
%Y A373673 - length A110969 (firsts A373669, sorted A373670)
%Y A373673 - min A373676
%Y A373673 - max A373677
%Y A373673 - sum A373678
%Y A373673 For antiruns of prime-powers:
%Y A373673 - length A373671
%Y A373673 - min A120430
%Y A373673 - max A006549
%Y A373673 - sum A373576
%Y A373673 For antiruns of non-prime-powers:
%Y A373673 - length A373672
%Y A373673 - min A373575
%Y A373673 - max A255346
%Y A373673 - sum A373679
%Y A373673 A000961 lists all powers of primes (A246655 if not including 1).
%Y A373673 A025528 counts prime-powers up to n.
%Y A373673 A057820 gives first differences of consecutive prime-powers, gaps A093555.
%Y A373673 A361102 lists all non-prime-powers (A024619 if not including 1).
%Y A373673 Cf. A000040, A007674, A014963, A053806, A054265, A068781, A072284, A073051, A120992, A373400.
%K A373673 nonn
%O A373673 1,2
%A A373673 _Gus Wiseman_, Jun 15 2024